Cardinal Properties of the Space of Quasicontinuous Functions under Topology of Uniform Convergence on Compact Subsets
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arXiv
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| Format: | Preprint |
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2024
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| author | Bishnoi, Chander Mohan Mishra, Sanjay |
| author_facet | Bishnoi, Chander Mohan Mishra, Sanjay |
| contents | In this paper, we investigate various cardinal properties of the space $Q_{C}X$ of all real-valued quasicontinuous functions on the topological space $X$, under the topology of uniform convergence on compact subsets. It begins by examining the relationship between tightness and other properties in the context of the space $X$, highlighting results such as the alignment of tightness $Q_{C}X$ with the compact Lindelöf number of $X$ under Hausdorff conditions and the countable tightness of $Q_{C}X$ when $X$ is second countable. Further investigations reveal conditions for the tightness of $Q_{C}X$ relative to $k$-covers of $X$, as well as connections between density tightness, fan tightness, and other properties in Hausdorff spaces. Additionally, we discuss the implications of the Frechet-Urysohn property $Q_{C}X$ for open $k$-covers in Hausdorff spaces. We explore relationships between $Q_{C}X$'s tightness, the Frechet-Urysohn property, and the $σ$-compactness of locally compact Hausdorff spaces $X$. Furthermore, we examine the $k_{f}$-covering property and the existence of $k$-covers in the context of Whyburn spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_02209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cardinal Properties of the Space of Quasicontinuous Functions under Topology of Uniform Convergence on Compact Subsets Bishnoi, Chander Mohan Mishra, Sanjay General Topology 54C35, 54A25, 54C08, 54C30, and 54D10 In this paper, we investigate various cardinal properties of the space $Q_{C}X$ of all real-valued quasicontinuous functions on the topological space $X$, under the topology of uniform convergence on compact subsets. It begins by examining the relationship between tightness and other properties in the context of the space $X$, highlighting results such as the alignment of tightness $Q_{C}X$ with the compact Lindelöf number of $X$ under Hausdorff conditions and the countable tightness of $Q_{C}X$ when $X$ is second countable. Further investigations reveal conditions for the tightness of $Q_{C}X$ relative to $k$-covers of $X$, as well as connections between density tightness, fan tightness, and other properties in Hausdorff spaces. Additionally, we discuss the implications of the Frechet-Urysohn property $Q_{C}X$ for open $k$-covers in Hausdorff spaces. We explore relationships between $Q_{C}X$'s tightness, the Frechet-Urysohn property, and the $σ$-compactness of locally compact Hausdorff spaces $X$. Furthermore, we examine the $k_{f}$-covering property and the existence of $k$-covers in the context of Whyburn spaces. |
| title | Cardinal Properties of the Space of Quasicontinuous Functions under Topology of Uniform Convergence on Compact Subsets |
| topic | General Topology 54C35, 54A25, 54C08, 54C30, and 54D10 |
| url | https://arxiv.org/abs/2412.02209 |